Higher order sliding mode control based on integral sliding mode

Abstract

A higher order sliding mode control scheme for uncertain nonlinear systems is proposed in the present paper. It is shown that the problem is equivalent to the finite time stabilization of higher order input–output dynamics with bounded uncertainties $(r\in \mathbb{N})$. The controller uses integral sliding mode concept and contains two parts. A part achieves finite time stabilization of the higher order input–output dynamics without uncertainties. The other part rejects bounded uncertainties throughout the entire response of the system. As a result, a higher order sliding mode is established. The advantages of the method are that its implementation is easy, the time convergence is chosen in advance and the robustness is ensured. An illustrative example of a car control shows the applicability of the method.

Introduction

The sliding mode control (SMC) is a powerful method to control high-order nonlinear dynamic systems operating under uncertainty conditions (Utkin, 1977, Utkin, 1992, Utkin et al., 1999). The technique consists of two stages. First, a sliding surface, to which the controlled system trajectories must belong, is designed with accordance to some performance criterion. Then, a discontinuous control is designed to force the system state to reach the sliding surface such that a sliding mode occurs on this manifold. When sliding mode is realized, the system exhibits robustness properties with respect to parameter perturbations and external matched disturbances.

In spite of claimed robustness properties, high frequency oscillations of the state trajectories around the sliding manifold known as chattering phenomenon (Utkin, 1992, Utkin et al., 1999) are the major obstacles for the implementation of SMC in a wide range of applications. A number of methods have been proposed to reliably prevent chattering: among them, the boundary layer solution (Slotine, 1984), observer-based solution (Utkin et al., 1999) and higher order SMC (Bartolini et al., 1998, Emelyanov et al., 1993, Levant, 1993, Levant, 2001). The current papers result is based on this latter approach and its main idea can be described as follows. Let $s(x,t)$ ($x\in {\mathbb{R}}^n$ is the state variable, $t\in {\mathbb{R}}^{+}$ the time variable) be the sliding variable and $r\in \mathbb{N}$ the sliding order. The control forces to zero in finite time s and its $(r-1)$ first higher time derivatives by acting discontinuously on the $r$th time derivative of s. Keeping the main advantages of standard SMC, the chattering effect is eliminated and higher order precision is provided. In the case of “real” SMC (Levant, 1993), if $\tau$ is the sampling time, the error is $o(\tau )$ in the case of standard SMC (Furuta, 1990) whereas it is $o({\tau }^r)$ in the $r$th order SMC (Levant, 1993).

In the case of second order SMC ($r=2$), many works have given solutions. Several second order sliding mode algorithms are proposed in Bartolini et al. (1998), Bartolini, Ferrara, Usai, and Utkin (2000), Emelyanov et al. (1993), Levant (1993). Arbitrary-order sliding controllers for single-input single-output systems (SISO) with finite time convergence have been recently proposed in Levant, 2001, Levant, 2003, Levant, 2005, Laghrouche, Plestan, and Glumineau (2006). To our best knowledge, these works are the most complete published on the higher order sliding mode approach. In Wu, Yu, and Man (1998), the so-called “terminal sliding modes control” proposes a solution for arbitrary order finite time convergence sliding mode. A major drawback of this approach is that all trajectories have to start from a prescribed sector of state space in order to have finite control, which is restrictive in particular for practical applications. The algorithm in Levant (2001) allows the tracking of smooth signals by tuning only one “gain” parameter and from the knowledge of the relative degree of the output. However, there is no constructive condition for the gain tuning, which has to be chosen sufficiently large (Levant, 2001). The controller proposed in Laghrouche, Plestan et al. (2006) combines standard SMC with linear quadratic (LQ) one over a finite time interval with a fixed final state. The algorithm uses the relative degree of the system with respect to the sliding variable s and the bounds of uncertainties. It has several advantages: the upper bound of the convergence time is known and can be adjusted in advance, the condition on the gain implies that its tuning is constructive, and the structure of the controller is well-adapted to practical implementations (Laghrouche, Plestan, & Glumineau, 2004b; Laghrouche, Smaoui, Plestan, & Brun, 2006). However, two drawbacks appear in this approach. It ensures only a practical sliding mode establishment (only convergence in finite time to an arbitrarily small vicinity of the origin is ensured), and the reaching time is bounded but cannot be fixed exactly and in advance.

The aim of this paper is to present a new arbitrary-order sliding mode controller for uncertain SISO minimum-phase nonlinear systems which is an alternative of Laghrouche, Plestan et al. (2006) without its drawbacks. The main objective of this new approach is to propose a controller for which the implementation is easy, the convergence time is finite and well-known in advance and the robustness is ensured during the entire response of the system. The design uses the integral sliding mode concept (Utkin & Shi, 1996). Actually, the problem of the higher order SMC of SISO minimum-phase uncertain systems is formulated in input–output terms only (as in Levant, 2001) through the differentiation of the sliding variable s, and is equivalent to the finite time stabilization of an $r$th order input–output linear dynamics with bounded uncertainties. The control strategy presented in the sequel, whose basic idea has been introduced in Laghrouche, Plestan, and Glumineau (2004a), contains two parts: a part is discontinuous, forces the establishment of a sliding mode on the integral sliding manifold, and ensures the robustness with respect to bounded uncertainties, throughout the entire response of the system. The other part, which is obtained through optimal feedback control over finite time interval with fixed final states (Rekasius, 1964), is used to stabilize to zero in finite time the $r$th order input–output dynamics without uncertainties. In Laghrouche et al. (2004a), this part is based on an open-loop control. Its advantage is the easy implementation; however, the open-loop control law depends only on the initial state, is precomputed and is applied for $t\in [0,t_{\mathrm{F}}]$. If, for some reason, the state is perturbed off the predicted optimal trajectory, or if the initial state measurements are noised, the system will not be, in general, in the desired position at the final time. These facts are well-known drawbacks of open-loop control w.r.t. closed-loop scheme and imply the real interest to develop, if possible, a feedback solution, as made in the sequel.

The paper is organized as follows. Section 2 states the problem and hypotheses. Section 3 displays the design of the higher order SMC with an integral approach. Its application on an academic car control example is carried out in Section 4.